Explicit Form of Station-Keeping and Formation Flying Controller for Libration Point Orbits

Abstract

I. IntroductionRecently unstable orbits in the vicinity of the libration points, or of the so-called Lagrangian points [1,2], in the circular restricted three-body problem (CRTBP) have attracted much attention, and station-keeping on them has been studied by several authors [3][4][5][6][7][8][9]. For formation flying along unstable orbits, a simple feedback law was proposed by Scheeres [10] by stabilizing the unstable manifold and creating additional center manifolds. Based on the output regulation theory of a linear system [11], a control law to realize the formation flying and station-keeping was proposed by Bando and Ichikawa [12]. The periodic Riccati differential equations were used by Peng et al. for the station-keeping and formation flying based on the linear periodic time-varying equation of the relative motion around a libration point orbit [13].In this paper, station-keeping and formation flying along unstable libration point orbits are considered.Using the nonlinear output regulation theory [14,15], a general form of controller is analytically derived to achieve station-keeping and formation flying of periodic and quasi-periodic (invariant tori) orbits embedded on a four-dimensional center manifold in the CRTBP.A standard approach to derive such a controller is to linearize the system along a reference orbit and then stabilize the linearized system [6,16]. In our approach, the problem is solved as a nonlinear output regulation problem which allows us to derive a general form for station-keeping and formation flying controllers without the linearization assumption. However, the nonlinear output regulation problem cannot be applied directly to this problem due to an unstable mode. Therefore, by using the center manifold theory in the formulation

Formation flying along libration point orbits using chattering attenuation sliding mode control

“…The state space form of nonlinear equations in Eq. 3 is then semilinear and is given by (Akiyama et al, 2018)…”

Station-keeping and formation flying based on nonlinear output regulation theory

“…Station-keeping and formation flying in the CRTBP have been studied by many authors [15,16,17,18,19,20,21,22,23,24]. Recently, based on the output regulation theory of a linear system [25], a control law to realize formation flying and station-keeping was proposed [26,27]. However, they utilized the dynamics obtained via feedback linearization, and therefore the control costs for large reference orbits become high due to the nonlinearity.…”

“…However, they utilized the dynamics obtained via feedback linearization, and therefore the control costs for large reference orbits become high due to the nonlinearity. The purpose of this paper is to propose the station-keeping and formation flying controllers based on the nonlinear output regulation theory [28,29] as the extension of the works [26,27] to deal with the relative motion with respect to a reference trajectory. First, a Fourier series approximation is introduced to describe a periodic or quasi-periodic orbit in the CRTBP.…”